Saddle-point resonances in few-body systems

Volume 177, number 3

22 February 1991

CHEMICAL PHYSICS LETTERS

Saddle-point resonances in few-body systems J.M. Rost and J.S. Briggs Fokultiitfir Physik, Albert-Ludwigs-Universitiit, Hermann-Herder-Strasse3, W-7800Freiburg, Germany Received 19 November 1990

The generality of resonance formation due to localisation on saddle points of a many-dimensional potential surface governing the motion of few-body systems is demonstrated. Adiabatic potential curves show sequences of avoided crossings. It is shown how diabatic potentials connecting these crossings can be constructed corresponding to preservation of motion on the saddle and leading to a higher degree of fragmentation of the complex. The method is illustrated by calculation of vibrational resonances in type ABA molecules and doubly excited electronic resonances in two-electron atoms.

In many problems of few-body dynamics, motion is governed in localised regions of configuration space by attractive potential wells of well-defined form. Where several such regions exist on a potential surface, they are connected by saddle or ridge structures delineating the boundaries between different regions. Within the localised regions, the motion is often of some simply soluble form. A common theoretical approach to such many-dimensional problems is to isolate certain coordinates, here denoted by T, which serve as adiabatic parameters to be fixed whilst the motion in all other dimensions is determined. The topology of the potential surface in the remaining variables (whose behaviour will be called the “inner” motion) then depends parametrically upon these adiabatic variables (whose behaviour will

be called the “outer” motion). Typically the number and nature of the potential craters and valleys and the connecting saddles or ridges varies as a function of these parameters, i.e. different types of localised motion are possible in different regions of the adiabatic parameter space. In quantum mechanics, the potential surface for the motion of the adiabatic coordinate itself is then determined by averaging over the “inner” motion, i.e. by constructing the adiabatic potentials as the continuous eigenenergies of the inner motion as a function of the adiabatic parameters. The presence of saddles between different localised regions of the inner motion manifests itself in the adiabatic curves for outer motion as sequences of avoided crossings as a function of r (see fig. 1d) .

co/Y*

p (m.4.u)

P

by)

Fig. 1. The double Morse potential V(p, cp) (a) forp= 1, (b) forp=p,=3.142, (c) forp=lO. The point q=O corresponds to r,=r,. (d) The series of A= + I adiabatic ABA potentials, (e) same as (d) but including the diabatic potential (-) and the static potential V(p, p=O) (-) on the saddle. 0009-26I4/9 l/S 03.50 6 1991 - Elsevier Science Publishers B.V. ( North-Holland )

321

Volume 177, number 3

CHEMICAL PHYSICS LETTERS

Sequences occur since adiabatic eigenenergies corresponding to increasing degrees of inner excitation show sharp changes of slope at increasing values of a given adiabatic parameter. These sharp changes of slope correspond to sharp changes in the localisation character of the inner motion and the avoided crossing is typified by a large amplitude for inner motion on the saddle connecting the two regions. These saddle regions which, because of their topology, involve unstable motion in at least one inner variable are of vital importance for the formation of resonances in few-body systems. The typical situation is that for a given value of z, a potential well confines part of the inner motion at a given tbtal system energy, but as the adiabatic parameter increases, this well splits into two. If the inner motion remains confined in one or the other well, then the adiabatic potential is followed and this leads to dissociation. However, if the inner motion remains stranded on the saddle as the two wells separate, the system remains in an unstable bound state, i.e. a resonance is formed. This stranding of the inner motion on the saddle occurs preferentially for fast change of 2; i.e. for diabatic motion. It is also apparent that in the crossing region, the two types of motion are almost degenerate; the adiabatic behaviour corresponds to a high degree of inner excitation (in the potential well) and low outer excitation corresponding to the thnshold for a particular dissociation channel. The diabatic behaviour corresponds to a low inner excitation (on the saddle) compensated by a high degree of outer excitation. This vibrational outer motion stabilises the unstable inner motion. The vibrational motion occurs in the diabatic potential well whose outer rim crosses the avoided crossings between the sequence of adiabatic curves, and is asymptotic to a threshold for more complete fragmentation of the few-body system. The importance of such inner motion on potential saddles has been recognised by Fano [ 1 ] who described avoided crossings between adiabatic curves and the crossing behaviour corresponding to propagation of the inner motion along the potential ridge. However, no prescription was given for the construction of a suitable diabatic state describing these crossings or for the mechanism of energy exchange between inner and outer motion which leads to stabilisation of locally unstable motion on the convex 322

22February1991

surface forming the ridge. Here, we provide a simple

method of constructing a diabatic state by a consideration of the nature of the internal motion at the symmetry-breaking or bifurcation value ts, where the transition from one-well to two-well structure occurs for fixed total system energy. Moreover, the universality of the phenomenon is demonstrated by application to resonance formation in two completely different three-body systems involving different forms of potential, mass ratios and non-separability. The basic problem can be expressed in the following way: The Schtidinger equation can be represented as (K+ V-E) y/=0,

(1)

where K is the kinetic energy operator of an N-body system involving 3N- 3 suitable mass-scaled coordinates and Yis the total potential energy of the system. We formulate a qua¶ble ansatz that splits off the coordinates r in such a way that the kinetic energy can be decomposed into K=&+K,(r)

(2)

.

The adiabatic approach solves the inner problem, only parametrically dependent upon 7, in the remaining coordinates denoted by the vector a, i.e. H,(z) =Kn(z) t V(z; a) .

(3)

The adiabatic approximation to the solution of ( 1) is a single-channel wavefunction of the form %‘,,(5 0) =~cY;,B(~)Q&Y(z; 0) I

(4)

where Qs satisfies the “inner” equation, HO@B(f,a) = &(+$(r,

0) ,

(5)

and &j the “outer” equation, [K+ U/?(c)IF,,(z) =&4FLy/?(~) .

(‘3)

The adiabatic potentials Ufi(z) are given by ~,(~)=~~(~)+(~BIH-~oI~B)a,

(7)

with Qs normalised as ( Oal Qa>,= 1. The first problem we consider is the vibration of molecules of the type ABA in a fixed bending-angle approximation. This system has only two degrees of freedom but has been much studied as a prototype of real molecules and accurate direct diagonalisation results exist for vibrational levels [ 2,3]. The model

Volume 177, number 3

CHEMICAL PHYSICS LETTER!?

is that of two identical Morse oscillators representing the AB interaction coupled by a mass polarisation term. If rA&rB* are appropriate relative separations forming Cartesian coordinates (x, y), then it is convenient [ 41 to transform to polar (hyperspherical) coordinates (p, ~0).With the hyperspherical radius p as adiabatic coordinate r and the pseudo-angle 8 as inner coordinate a, the problem is exactly of the form expressed by eqs. ( 1)-( 7 ). The inner potential V(P), parametrically dependent upon the adiabatic parameter p, is shown in figs. la-l c. For small values of p, the potential has a single attractive well. Asp increases, a bifurcation point pSis reached at which a2V//aq21pr=o=0. Forp>p,, in this case of one-dimensional inner motion, it is simply the convex potential surface V(~“0) which forms the “saddle” giving rise to locally unstable saddle motion. The sequence shown in fig. 1 illustrates the transition from one- to two-well inner motion [ 51 discussed above. Eq. (5 ) has been solved and the adiabatic potentials constructed according to eq. (7 ). They can be classified into two groups [ I], depending upon whether A = ( - 1) p= -t-1, corresponding to the presence of an anti-node or node at the saddle point (maximum of V(p) for p>pS). Our aim is to construct the single diabatic curve connecting the avoided crossings (see fig. Id) and asymptotic to the three-body break-up threshold. To this end, we adopt a strategy based on physical reasoning. If the inner motion is to remain stranded on the saddle as p increases beyond pS,it is clear that the form of the inner wavefunction must be frozen at this point and maintain this form with large amplitude on the saddle as p-+03. It is also clear that since the diabatic curve connects all members of the adiabatic se quence, beginning with the lowest state of a given symmetry, it is this lowest-state wavefunction that is relevant. Hence, for A= + 1, the diabatic potential replacing the adiabatic U,(p) in eq. (6) is constructed as the expectation value of the total Hamiltonian, LTD(P)=(~~(VI,PS)IH(P,Q)I~~((Q),PS))(P.

(8)

Although the form of &(pl, pS) is fixed by numerical solution of (5 ) at pS,to obtain high numerical accuracy in the energies it is necessary, in the sense of a trial function, to include one variational

22February 1991

parameter which scales the hyperradius p. The lowest three levels lie below the first two-body threshold and, hence, are bound. Therefore, the parameter has been chosen by fitting the lowest diabatic energies to the corresponding levels from the adiabatic approximation (6) which is known to be accurate for these states [ 61. When this is done, the resulting diabatic energies of all the higher-lying resonances are in close agreement with the large basis-set calculations [ 21 over the complete energy range up to the three-body break-up threshold (see table 1). The accuracy of the resonance energies calculated in the simple diabatic potential illustrates the fundamental features of saddle dynamics. The corresponding resonances in the adiabatic curves are the lowest vibrational outer states built on states of increasing inner excitation, i.e. the states with wavefunctions !P&, I) =F,B(p)Os(~; p). By contrast, the diabatic resonances are the states of increasing outer vibrational excitation built on the state of lowest inner or saddle excitation, i.e. the states with wavefunctions !P&(p,rj9)=F&)GO(yr;p). The neardegeneracy of these two sets of independent modes facilitates the exchange of excitation energy (from adiabatic inner saddle motion to diabatic outer vibrational motion) that is characteristic for saddle resonances. The diabatic potential follows closely the static potential V(p, q=O) on the saddle which is of Morse form (fig. 1e). That is, the resonance energies themTable 1 Energy levels for the ABA vibrational motion classified by the number (Yof zeros in the diabatic eigenfunction.

The blank line

separates bound states from resonances (Y

-E (eV) diabatic

ref. [2]

Morse

0 2

1.353 1.075

1.370 1.086

1.352

4

0.829

0.835

1.076 0.832

0.617 0.436 0.288 0.171 0.0851 0.0293 0.00259

0.620 0.439 0.292 0.167 0.0843 0.0294 0.00258

0.619 0.438 0.288 0.169 0.0822 0.0268 0.0028

6 8 10 12 14 16 18

323

Volume177,number3

CHEMICAL PHYSICS LETTERS

selves can be fitted by a Morse-type formula [ 71 as can be seen from table 1. A similar feature will emerge below for the two-electron, one-nucleus problem, which will now be considered in more detail. In the MO model of this system, the interelectronic separation R plays the role of the adiabatic coordinates and the coordinate r of the electronic center-of-mass relative to the nucleus is the inner coordinate o. For simplicity, only states of total S symmetry will be considered so that the outer R-motion is one-dimensional. Furthermore, cylindrical symmetry reduces the inner problem to a two-dimensional one. The potential surface for inner motion, shown in fig. 2a, now has a true saddle point since motion perpendicular to the R axis is stable. For the pure Coulomb adiabatic problem, the inner motion described by (5) is separable in prolate spheroidal coordinates A,p, I, where A= (r, + r2) /R, p= (r, -r2) /R and r,, r2 are the electron-nucleus vectors. Now j7= (n,, np m) denotes the set of quantum numbers corresponding, respectively, to the number of ellipsoidal, hyperboloidal and azimuthal nodal surfaces which are conserved for all R. In these coordinates, the unstable part of the saddle-point motion is described by the p coordinate and the saddle point is at r, =r,=R/2, i.e. A= 1, p=O. Again the adiabatic potential curves can be ordered into saddle sequences [ 8 ] with given n,, m and n, even or odd (A= ? 1). Such sequences are character&d by avoided crossings, where the inner motion changes from sitting on the saddle to being localised in one or the other well. The sequence (0, +even, 0) is shown in fig. 2b. The similarity of fig. Id, for the vibrations of an ABA molecule, and fig. 2b, for the electronic states of two-electron atoms, is clear. The diabatic curve connecting the avoided crossings of fig. 2b has been constructed previously [ 9 1 using arguments based on the form of the eigenstates of the operator H- l/R. Here, we show that this is an example of the general strategy of “freezing” the form of the inner wavefunction at the bifurcation point of the inner potential. The key point is that for singular Coulomb potentials, the bifurcation point is at R=O. A consideration of the inner wavefunction in the bifurcation limit R-0 gives Gmw exp( -ZRA). The use of this wavefunction form in (5) and (7) generates the diabatic potential shown in fig. 2c. Again, to achieve high numerical accuracy, 324

22February 1991

0-c

0.00

200.00

400.00

RZ (au) Fig. 2. (a) The two-centre Coulomb potential for R=2 au. The inner coordinates are parallel (x) and perpendicular (z) to 8. (b) The lowest A = t 1 series of IS adiabatic two-electron potentials, (c) same as (b) but including the diabatic potential. The energy is expressed as U(R)= - 1/2N&(R).

was replaced by an effective charge in a trial-function sense and the optimum value decided by minimizing the ground-state energy which represents the only symmetric bound state in the diabatic potential. High-lying ‘SCresonance energies calculated as the vibrational energies of the single diabatic potential with A = + 1, then agree extremely closely (see table 2) with resonance energies obtained by direct diagonalisation of the full problem in basis sets of 2

Volume 177, number 3

CHEMICAL PHYSICS LETTERS

Table 2 Same as table I but for ‘SCHe levels. The relation between orand the principal electronic quantum number N is given by cx= 2(N-1) -E (au)

a

0 2 4 6 8 10 12 14 16 18

diabatic

ref. [ 101

2.891

2.903

0.773 Q.353 0.201 0.130 0.0908 0.0670 0.05 14 0.0407 0.033 1

0.778 0.354 0.201 0.129 0.0903

Rydberg

0.772 0.353 0.201 0.130 0.0908 0.0670 0.0514 0.0407 0.0331

functions spanning the (rr, r2) = (I, R) space [101. This illustrates once more the formation of saddle resonances by exchange of saddle excitation from the set of adiabatic F,,,(R)$(r, R) vibrational ground states to the set of diabatic vibrationally excited F&R)@,,(r, R) saddle ground states. In the case of the Coulomb potential, the diabatic potential can be obtained in closed form and has a long-range Coulomb form supporting a Rydberg series of doubly-excited resonances. That explains why high-lying states can be fitted to a double Rydberg formula [ 111.The resonance positions obtained from such a formula are also shown in table 2. To summarise, we have indicated the generality of the formation of resonances by localisation of inner motion to a saddle point. In the adiabatic treatment, sequences of potential curves, characterised by saddle symmetry A = + 1, show avoided crossings whose locus is approximately the energy of the saddle point as a function of the adiabatic distance. It has been shown how single diabatic potentials connecting these avoided crossings can be constructed from a knowledge of the form of the inner wavefunction at the bifurcation point of the inner potential. The formation of the resonances on the saddle is interpreted as due to a quasi-resonant exchange of energy between adiabatic inner motion and diabatic outer vibrational motion [ 91. A similar quasi-degeneracy of the asymmetric (adiabatic) and symmetric (diabatic) stretch modes in ABA molecules has been established [ 3 1. The diabatic energy levels calculated here agree

22 February 199I

with the results of large basis-set direct-diagonalisation calculations for helium [ IO] and ABA molecules [ 21 in so far as they exist (see tables 1 and 2). In each case+however, the diabatic potential has a well-defined asymptotic form so that energy levels all the way up to the three-body threshold can easily be calculated. Furthermore, this form explains why the resonances can be fitted to a Morse series (molecule, table 1) or a Rydberg series (atom, table 2). For low to intermediate quantum numbers, hyperspherical adiabatic calculations have been performed for helium [ 121, H- [ 131 and ABA mole cules [ 61. These energies are also in close agreement with directdiagonalisation results. This perhaps surprising agreement of diabatic and adiabatic methods is not completely understood but the key lies in the approximate degeneracy of asymmetric and symmetric stretch modes evident in both systems referred to above. This may turn out to be a generic property of saddle resonances. This also implies that neither type of motion can be fully correct. Hence, the more exact wavefunctions that are necessary to describe properties of the resonant states more sensitive than mere energy positions, e.g. their widths and transition matrix elements to bound states, will appear as linear combinations of the extreme limits of completely adiabatic or diabatic motion. This work has been supported by the DFG in SFB276 TPAl. References [ I] U. Fano, Phys. Rev. A 22 ( 1980) 2660. [ 2 ] R.H. Bissling, R. Kosloff, J. Manz. F. Mrugala, J. Romelt and G. Weichselbaumcr, J. Chem. Phys. 86 (1987) 2626.

[ 3 ] KC. Kulander, J. Manz and H.H.R. Schor, J. Chem. Phys. 82 (1985) 3088. [4] J. Manz, Commun. At. Mol. Phys. 17 (1985) 91. [ 5 ] V. Aquilanti, G. Grossi and A. Lagana. Chem. Phys. Letters 93 (1982) 174. [6] J. Manz and H.H.R. Schor, Chem. Phys. Letters 107 (1984) 542. [ 7 ] S. Fliigge,Practical quantum mechanics (Springer, Berlin, 1977). [8] J.M. Rost and J.S. Briggs, J. Phys. B 23 (1990) L339. [9] J.M. Rost and J.S. Briggs, J. Phys. B 22 (1989) 3587. [IO] Y.K. Ho, Phys. Rev. A 34 (1986) 4402. [ 1I ] F.H. Read, J. Phys. B 10 ( 1976) 449. [ 121N. Koyama, A. Taknfuij and M. Matsuzawa, J. Phys. B 22 (1989) 553. [ 131H.R. Sadeghpour and C.H. Greene, Phys. Rev. Letters 65 (1990) 313.

325